Bounded and Divergent Trajectories And Expanding Curves on Homogeneous Spaces

TL;DR

This paper investigates the Hausdorff dimension of bounded and divergent orbits on homogeneous spaces, providing sharp bounds and applications to Diophantine approximation, especially for curves tangent to certain subgroups.

Contribution

It introduces new bounds on orbit dimensions for specific curves on homogeneous spaces and applies these results to problems in Diophantine approximation and linear forms.

Findings

Bounded orbits have full Hausdorff dimension and are winning.

Divergent on average orbits have sharp upper bounds on Hausdorff dimension.

Dimension of singular points in linear forms space is at most 1/2.

Abstract

Suppose $g_t$ is a $1$-parameter $\mathrm{Ad}$-diagonalizable subgroup of a Lie group $G$ and $\Gamma < G$ is a lattice. We study the dimension of bounded and divergent orbits of $g_t$ emanating from a class of curves lying on leaves of the unstable foliation of $g_t$ on the homogeneous space $G/\Gamma$. We obtain sharp upper bounds on the Hausdorff dimension of divergent on average orbits and show that the set of bounded orbits is winning in the sense of Schmidt (and, hence, has full dimension). The class of curves we study is roughly characterized by being tangent to copies of $\mathrm{SL}(2,\mathbb{R})$ inside $G$, which are not contained in a proper parabolic subgroup of $G$. We describe applications of our results to problems in Diophantine approximation by number fields and intrinsic Diophantine approximation on spheres. Our methods also yield the following result for lines in the space of square systems of linear forms: suppose $\varphi(s) = sY + Z$ where $Y\in \mathrm{GL}(n,\mathbb{R})$ and $Z\in M_{n,n}(\mathbb{R})$. Then, the dimension of the set of points $s$ such that $\varphi(s)$ is singular is at most $1/2$ while badly approximable points have Hausdorff dimension equal to $1$.